theory Spec

imports Base Main

begin

chapter {* Specifications *}

text {*

  The Isabelle/Isar theory format integrates specifications and

  proofs, supporting interactive development with unlimited undo

  operation.  There is an integrated document preparation system (see

  \chref{ch:document-prep}), for typesetting formal developments

  together with informal text.  The resulting hyper-linked PDF

  documents can be used both for WWW presentation and printed copies.

  The Isar proof language (see \chref{ch:proofs}) is embedded into the

  theory language as a proper sub-language.  Proof mode is entered by

  stating some @{command theorem} or @{command lemma} at the theory

  level, and left again with the final conclusion (e.g.\ via @{command

  qed}).  Some theory specification mechanisms also require a proof,

  such as @{command typedef} in HOL, which demands non-emptiness of

  the representing sets.

*}

section {* Defining theories \label{sec:begin-thy} *}

text {*

  \begin{matharray}{rcl}

    @{command_def "theory"} & : & @{text "toplevel \<rightarrow> theory"} \\

    @{command_def (global) "end"} & : & @{text "theory \<rightarrow> toplevel"} \\

  \end{matharray}

  Isabelle/Isar theories are defined via theory files, which may

  contain both specifications and proofs; occasionally definitional

  mechanisms also require some explicit proof.  The theory body may be

  sub-structured by means of \emph{local theory targets}, such as

  @{command "locale"} and @{command "class"}.

  The first proper command of a theory is @{command "theory"}, which

  indicates imports of previous theories and optional dependencies on

  other source files (usually in ML).  Just preceding the initial

  @{command "theory"} command there may be an optional @{command

  "header"} declaration, which is only relevant to document

  preparation: see also the other section markup commands in

  \secref{sec:markup}.

  A theory is concluded by a final @{command (global) "end"} command,

  one that does not belong to a local theory target.  No further

  commands may follow such a global @{command (global) "end"},

  although some user-interfaces might pretend that trailing input is

  admissible.

  @{rail "

    @@{command theory} @{syntax name} \\ @'imports' (@{syntax name} +) uses? @'begin'

    ;

    uses: @'uses' ((@{syntax name} | @{syntax parname}) +)

  "}

  \begin{description}

  \item @{command "theory"}~@{text "A \<IMPORTS> B\<^sub>1 \<dots> B\<^sub>n \<BEGIN>"}

  starts a new theory @{text A} based on the merge of existing

  theories @{text "B\<^sub>1 \<dots> B\<^sub>n"}.

  Due to the possibility to import more than one ancestor, the

  resulting theory structure of an Isabelle session forms a directed

  acyclic graph (DAG).  Isabelle's theory loader ensures that the

  sources contributing to the development graph are always up-to-date:

  changed files are automatically reloaded whenever a theory header

  specification is processed.

  The optional @{keyword_def "uses"} specification declares additional

  dependencies on extra files (usually ML sources).  Files will be

  loaded immediately (as ML), unless the name is parenthesized.  The

  latter case records a dependency that needs to be resolved later in

  the text, usually via explicit @{command_ref "use"} for ML files;

  other file formats require specific load commands defined by the

  corresponding tools or packages.

  \item @{command (global) "end"} concludes the current theory

  definition.  Note that local theory targets involve a local

  @{command (local) "end"}, which is clear from the nesting.

  \end{description}

*}

section {* Local theory targets \label{sec:target} *}

text {*

  A local theory target is a context managed separately within the

  enclosing theory.  Contexts may introduce parameters (fixed

  variables) and assumptions (hypotheses).  Definitions and theorems

  depending on the context may be added incrementally later on.  Named

  contexts refer to locales (cf.\ \secref{sec:locale}) or type classes

  (cf.\ \secref{sec:class}); the name ``@{text "-"}'' signifies the

  global theory context.

  \begin{matharray}{rcll}

    @{command_def "context"} & : & @{text "theory \<rightarrow> local_theory"} \\

    @{command_def (local) "end"} & : & @{text "local_theory \<rightarrow> theory"} \\

  \end{matharray}

  @{rail "

    @@{command context} @{syntax nameref} @'begin'

    ;

    @{syntax_def target}: '(' @'in' @{syntax nameref} ')'

  "}

  \begin{description}

  \item @{command "context"}~@{text "c \<BEGIN>"} recommences an

  existing locale or class context @{text c}.  Note that locale and

  class definitions allow to include the @{keyword "begin"} keyword as

  well, in order to continue the local theory immediately after the

  initial specification.

  \item @{command (local) "end"} concludes the current local theory

  and continues the enclosing global theory.  Note that a global

  @{command (global) "end"} has a different meaning: it concludes the

  theory itself (\secref{sec:begin-thy}).

  \item @{text "("}@{keyword_def "in"}~@{text "c)"} given after any

  local theory command specifies an immediate target, e.g.\

  ``@{command "definition"}~@{text "(\<IN> c) \<dots>"}'' or ``@{command

  "theorem"}~@{text "(\<IN> c) \<dots>"}''.  This works both in a local or

  global theory context; the current target context will be suspended

  for this command only.  Note that ``@{text "(\<IN> -)"}'' will

  always produce a global result independently of the current target

  context.

  \end{description}

  The exact meaning of results produced within a local theory context

  depends on the underlying target infrastructure (locale, type class

  etc.).  The general idea is as follows, considering a context named

  @{text c} with parameter @{text x} and assumption @{text "A[x]"}.

  Definitions are exported by introducing a global version with

  additional arguments; a syntactic abbreviation links the long form

  with the abstract version of the target context.  For example,

  @{text "a \<equiv> t[x]"} becomes @{text "c.a ?x \<equiv> t[?x]"} at the theory

  level (for arbitrary @{text "?x"}), together with a local

  abbreviation @{text "c \<equiv> c.a x"} in the target context (for the

  fixed parameter @{text x}).

  Theorems are exported by discharging the assumptions and

  generalizing the parameters of the context.  For example, @{text "a:

  B[x]"} becomes @{text "c.a: A[?x] \<Longrightarrow> B[?x]"}, again for arbitrary

  @{text "?x"}.

*}

section {* Basic specification elements *}

text {*

  \begin{matharray}{rcll}

    @{command_def "axiomatization"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\

    @{command_def "definition"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{attribute_def "defn"} & : & @{text attribute} \\

    @{command_def "abbreviation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "print_abbrevs"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow> "} \\

  \end{matharray}

  These specification mechanisms provide a slightly more abstract view

  than the underlying primitives of @{command "consts"}, @{command

  "defs"} (see \secref{sec:consts}), and @{command "axioms"} (see

  \secref{sec:axms-thms}).  In particular, type-inference is commonly

  available, and result names need not be given.

  @{rail "

    @@{command axiomatization} @{syntax \"fixes\"}? (@'where' specs)?

    ;

    @@{command definition} @{syntax target}? \\

      (decl @'where')? @{syntax thmdecl}? @{syntax prop}

    ;

    @@{command abbreviation} @{syntax target}? @{syntax mode}? \\

      (decl @'where')? @{syntax prop}

    ;

    @{syntax_def \"fixes\"}: ((@{syntax name} ('::' @{syntax type})?

      @{syntax mixfix}? | @{syntax vars}) + @'and')

    ;

    specs: (@{syntax thmdecl}? @{syntax props} + @'and')

    ;

    decl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}?

  "}

  \begin{description}

  \item @{command "axiomatization"}~@{text "c\<^sub>1 \<dots> c\<^sub>m \<WHERE> \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}

  introduces several constants simultaneously and states axiomatic

  properties for these.  The constants are marked as being specified

  once and for all, which prevents additional specifications being

  issued later on.

  Note that axiomatic specifications are only appropriate when

  declaring a new logical system; axiomatic specifications are

  restricted to global theory contexts.  Normal applications should

  only use definitional mechanisms!

  \item @{command "definition"}~@{text "c \<WHERE> eq"} produces an

  internal definition @{text "c \<equiv> t"} according to the specification

  given as @{text eq}, which is then turned into a proven fact.  The

  given proposition may deviate from internal meta-level equality

  according to the rewrite rules declared as @{attribute defn} by the

  object-logic.  This usually covers object-level equality @{text "x =

  y"} and equivalence @{text "A \<leftrightarrow> B"}.  End-users normally need not

  change the @{attribute defn} setup.

  Definitions may be presented with explicit arguments on the LHS, as

  well as additional conditions, e.g.\ @{text "f x y = t"} instead of

  @{text "f \<equiv> \<lambda>x y. t"} and @{text "y \<noteq> 0 \<Longrightarrow> g x y = u"} instead of an

  unrestricted @{text "g \<equiv> \<lambda>x y. u"}.

  \item @{command "abbreviation"}~@{text "c \<WHERE> eq"} introduces a

  syntactic constant which is associated with a certain term according

  to the meta-level equality @{text eq}.

  Abbreviations participate in the usual type-inference process, but

  are expanded before the logic ever sees them.  Pretty printing of

  terms involves higher-order rewriting with rules stemming from

  reverted abbreviations.  This needs some care to avoid overlapping

  or looping syntactic replacements!

  The optional @{text mode} specification restricts output to a

  particular print mode; using ``@{text input}'' here achieves the

  effect of one-way abbreviations.  The mode may also include an

  ``@{keyword "output"}'' qualifier that affects the concrete syntax

  declared for abbreviations, cf.\ @{command "syntax"} in

  \secref{sec:syn-trans}.

  \item @{command "print_abbrevs"} prints all constant abbreviations

  of the current context.

  \end{description}

*}

section {* Generic declarations *}

text {*

  Arbitrary operations on the background context may be wrapped-up as

  generic declaration elements.  Since the underlying concept of local

  theories may be subject to later re-interpretation, there is an

  additional dependency on a morphism that tells the difference of the

  original declaration context wrt.\ the application context

  encountered later on.  A fact declaration is an important special

  case: it consists of a theorem which is applied to the context by

  means of an attribute.

  \begin{matharray}{rcl}

    @{command_def "declaration"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "syntax_declaration"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "declare"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

  \end{matharray}

  @{rail "

    (@@{command declaration} | @@{command syntax_declaration})

      ('(' @'pervasive' ')')? \\ @{syntax target}? @{syntax text}

    ;

    @@{command declare} @{syntax target}? (@{syntax thmrefs} + @'and')

  "}

  \begin{description}

  \item @{command "declaration"}~@{text d} adds the declaration

  function @{text d} of ML type @{ML_type declaration}, to the current

  local theory under construction.  In later application contexts, the

  function is transformed according to the morphisms being involved in

  the interpretation hierarchy.

  If the @{text "(pervasive)"} option is given, the corresponding

  declaration is applied to all possible contexts involved, including

  the global background theory.

  \item @{command "syntax_declaration"} is similar to @{command

  "declaration"}, but is meant to affect only ``syntactic'' tools by

  convention (such as notation and type-checking information).

  \item @{command "declare"}~@{text thms} declares theorems to the

  current local theory context.  No theorem binding is involved here,

  unlike @{command "theorems"} or @{command "lemmas"} (cf.\

  \secref{sec:axms-thms}), so @{command "declare"} only has the effect

  of applying attributes as included in the theorem specification.

  \end{description}

*}

section {* Locales \label{sec:locale} *}

text {*

  Locales are parametric named local contexts, consisting of a list of

  declaration elements that are modeled after the Isar proof context

  commands (cf.\ \secref{sec:proof-context}).

*}

subsection {* Locale expressions \label{sec:locale-expr} *}

text {*

  A \emph{locale expression} denotes a structured context composed of

  instances of existing locales.  The context consists of a list of

  instances of declaration elements from the locales.  Two locale

  instances are equal if they are of the same locale and the

  parameters are instantiated with equivalent terms.  Declaration

  elements from equal instances are never repeated, thus avoiding

  duplicate declarations.

  @{rail "

    @{syntax_def locale_expr}: (instance + '+') (@'for' (@{syntax \"fixes\"} + @'and'))?

    ;

    instance: (qualifier ':')? @{syntax nameref} (pos_insts | named_insts)

    ;

    qualifier: @{syntax name} ('?' | '!')?

    ;

    pos_insts: ('_' | @{syntax term})*

    ;

    named_insts: @'where' (@{syntax name} '=' @{syntax term} + @'and')

  "}

  A locale instance consists of a reference to a locale and either

  positional or named parameter instantiations.  Identical

  instantiations (that is, those that instante a parameter by itself)

  may be omitted.  The notation `@{text "_"}' enables to omit the

  instantiation for a parameter inside a positional instantiation.

  Terms in instantiations are from the context the locale expressions

  is declared in.  Local names may be added to this context with the

  optional for clause.  In addition, syntax declarations from one

  instance are effective when parsing subsequent instances of the same

  expression.

  Instances have an optional qualifier which applies to names in

  declarations.  Names include local definitions and theorem names.

  If present, the qualifier itself is either optional

  (``\texttt{?}''), which means that it may be omitted on input of the

  qualified name, or mandatory (``\texttt{!}'').  If neither

  ``\texttt{?}'' nor ``\texttt{!}'' are present, the command's default

  is used.  For @{command "interpretation"} and @{command "interpret"}

  the default is ``mandatory'', for @{command "locale"} and @{command

  "sublocale"} the default is ``optional''.

*}

subsection {* Locale declarations *}

text {*

  \begin{matharray}{rcl}

    @{command_def "locale"} & : & @{text "theory \<rightarrow> local_theory"} \\

    @{command_def "print_locale"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

    @{command_def "print_locales"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

    @{method_def intro_locales} & : & @{text method} \\

    @{method_def unfold_locales} & : & @{text method} \\

  \end{matharray}

  \indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}

  \indexisarelem{defines}\indexisarelem{notes}

  @{rail "

    @@{command locale} @{syntax name} ('=' @{syntax locale})? @'begin'?

    ;

    @@{command print_locale} '!'? @{syntax nameref}

    ;

    @{syntax_def locale}: @{syntax context_elem}+ |

      @{syntax locale_expr} ('+' (@{syntax context_elem}+))?

    ;

    @{syntax_def context_elem}:

      @'fixes' (@{syntax \"fixes\"} + @'and') |

      @'constrains' (@{syntax name} '::' @{syntax type} + @'and') |

      @'assumes' (@{syntax props} + @'and') |

      @'defines' (@{syntax thmdecl}? @{syntax prop} @{syntax prop_pat}? + @'and') |

      @'notes' (@{syntax thmdef}? @{syntax thmrefs} + @'and')

  "}

  \begin{description}

  \item @{command "locale"}~@{text "loc = import + body"} defines a

  new locale @{text loc} as a context consisting of a certain view of

  existing locales (@{text import}) plus some additional elements

  (@{text body}).  Both @{text import} and @{text body} are optional;

  the degenerate form @{command "locale"}~@{text loc} defines an empty

  locale, which may still be useful to collect declarations of facts

  later on.  Type-inference on locale expressions automatically takes

  care of the most general typing that the combined context elements

  may acquire.

  The @{text import} consists of a structured locale expression; see

  \secref{sec:proof-context} above.  Its for clause defines the local

  parameters of the @{text import}.  In addition, locale parameters

  whose instantance is omitted automatically extend the (possibly

  empty) for clause: they are inserted at its beginning.  This means

  that these parameters may be referred to from within the expression

  and also in the subsequent context elements and provides a

  notational convenience for the inheritance of parameters in locale

  declarations.

  The @{text body} consists of context elements.

  \begin{description}

  \item @{element "fixes"}~@{text "x :: \<tau> (mx)"} declares a local

  parameter of type @{text \<tau>} and mixfix annotation @{text mx} (both

  are optional).  The special syntax declaration ``@{text

  "(\<STRUCTURE>)"}'' means that @{text x} may be referenced

  implicitly in this context.

  \item @{element "constrains"}~@{text "x :: \<tau>"} introduces a type

  constraint @{text \<tau>} on the local parameter @{text x}.  This

  element is deprecated.  The type constraint should be introduced in

  the for clause or the relevant @{element "fixes"} element.

  \item @{element "assumes"}~@{text "a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}

  introduces local premises, similar to @{command "assume"} within a

  proof (cf.\ \secref{sec:proof-context}).

  \item @{element "defines"}~@{text "a: x \<equiv> t"} defines a previously

  declared parameter.  This is similar to @{command "def"} within a

  proof (cf.\ \secref{sec:proof-context}), but @{element "defines"}

  takes an equational proposition instead of variable-term pair.  The

  left-hand side of the equation may have additional arguments, e.g.\

  ``@{element "defines"}~@{text "f x\<^sub>1 \<dots> x\<^sub>n \<equiv> t"}''.

  \item @{element "notes"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}

  reconsiders facts within a local context.  Most notably, this may

  include arbitrary declarations in any attribute specifications

  included here, e.g.\ a local @{attribute simp} rule.

  The initial @{text import} specification of a locale expression

  maintains a dynamic relation to the locales being referenced

  (benefiting from any later fact declarations in the obvious manner).

  \end{description}

  Note that ``@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}'' patterns given

  in the syntax of @{element "assumes"} and @{element "defines"} above

  are illegal in locale definitions.  In the long goal format of

  \secref{sec:goals}, term bindings may be included as expected,

  though.

  \medskip Locale specifications are ``closed up'' by

  turning the given text into a predicate definition @{text

  loc_axioms} and deriving the original assumptions as local lemmas

  (modulo local definitions).  The predicate statement covers only the

  newly specified assumptions, omitting the content of included locale

  expressions.  The full cumulative view is only provided on export,

  involving another predicate @{text loc} that refers to the complete

  specification text.

  In any case, the predicate arguments are those locale parameters

  that actually occur in the respective piece of text.  Also note that

  these predicates operate at the meta-level in theory, but the locale

  packages attempts to internalize statements according to the

  object-logic setup (e.g.\ replacing @{text \<And>} by @{text \<forall>}, and

  @{text "\<Longrightarrow>"} by @{text "\<longrightarrow>"} in HOL; see also

  \secref{sec:object-logic}).  Separate introduction rules @{text

  loc_axioms.intro} and @{text loc.intro} are provided as well.

  \item @{command "print_locale"}~@{text "locale"} prints the

  contents of the named locale.  The command omits @{element "notes"}

  elements by default.  Use @{command "print_locale"}@{text "!"} to

  have them included.

  \item @{command "print_locales"} prints the names of all locales

  of the current theory.

  \item @{method intro_locales} and @{method unfold_locales}

  repeatedly expand all introduction rules of locale predicates of the

  theory.  While @{method intro_locales} only applies the @{text

  loc.intro} introduction rules and therefore does not decend to

  assumptions, @{method unfold_locales} is more aggressive and applies

  @{text loc_axioms.intro} as well.  Both methods are aware of locale

  specifications entailed by the context, both from target statements,

  and from interpretations (see below).  New goals that are entailed

  by the current context are discharged automatically.

  \end{description}

*}

subsection {* Locale interpretations *}

text {*

  Locale expressions may be instantiated, and the instantiated facts

  added to the current context.  This requires a proof of the

  instantiated specification and is called \emph{locale

  interpretation}.  Interpretation is possible in locales @{command

  "sublocale"}, theories (command @{command "interpretation"}) and

  also within a proof body (command @{command "interpret"}).

  \begin{matharray}{rcl}

    @{command_def "interpretation"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

    @{command_def "interpret"} & : & @{text "proof(state) | proof(chain) \<rightarrow> proof(prove)"} \\

    @{command_def "sublocale"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

    @{command_def "print_dependencies"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

    @{command_def "print_interps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  \end{matharray}

  @{rail "

    @@{command interpretation} @{syntax locale_expr} equations?

    ;

    @@{command interpret} @{syntax locale_expr} equations?

    ;

    @@{command sublocale} @{syntax nameref} ('<' | '\<subseteq>') @{syntax locale_expr} \\

      equations?

    ;

    @@{command print_dependencies} '!'? @{syntax locale_expr}

    ;

    @@{command print_interps} @{syntax nameref}

    ;

    equations: @'where' (@{syntax thmdecl}? @{syntax prop} + @'and')

  "}

  \begin{description}

  \item @{command "interpretation"}~@{text "expr \<WHERE> eqns"}

  interprets @{text expr} in the theory.  The command generates proof

  obligations for the instantiated specifications (assumes and defines

  elements).  Once these are discharged by the user, instantiated

  facts are added to the theory in a post-processing phase.

  Additional equations, which are unfolded during

  post-processing, may be given after the keyword @{keyword "where"}.

  This is useful for interpreting concepts introduced through

  definitions.  The equations must be proved.

  The command is aware of interpretations already active in the

  theory, but does not simplify the goal automatically.  In order to

  simplify the proof obligations use methods @{method intro_locales}

  or @{method unfold_locales}.  Post-processing is not applied to

  facts of interpretations that are already active.  This avoids

  duplication of interpreted facts, in particular.  Note that, in the

  case of a locale with import, parts of the interpretation may

  already be active.  The command will only process facts for new

  parts.

  Adding facts to locales has the effect of adding interpreted facts

  to the theory for all interpretations as well.  That is,

  interpretations dynamically participate in any facts added to

  locales.  Note that if a theory inherits additional facts for a

  locale through one parent and an interpretation of that locale

  through another parent, the additional facts will not be

  interpreted.

  \item @{command "interpret"}~@{text "expr \<WHERE> eqns"} interprets

  @{text expr} in the proof context and is otherwise similar to

  interpretation in theories.  Note that rewrite rules given to

  @{command "interpret"} after the @{keyword "where"} keyword should be

  explicitly universally quantified.

  \item @{command "sublocale"}~@{text "name \<subseteq> expr \<WHERE>

  eqns"}

  interprets @{text expr} in the locale @{text name}.  A proof that

  the specification of @{text name} implies the specification of

  @{text expr} is required.  As in the localized version of the

  theorem command, the proof is in the context of @{text name}.  After

  the proof obligation has been discharged, the facts of @{text expr}

  become part of locale @{text name} as \emph{derived} context

  elements and are available when the context @{text name} is

  subsequently entered.  Note that, like import, this is dynamic:

  facts added to a locale part of @{text expr} after interpretation

  become also available in @{text name}.

  Only specification fragments of @{text expr} that are not already

  part of @{text name} (be it imported, derived or a derived fragment

  of the import) are considered in this process.  This enables

  circular interpretations provided that no infinite chains are

  generated in the locale hierarchy.

  If interpretations of @{text name} exist in the current theory, the

  command adds interpretations for @{text expr} as well, with the same

  qualifier, although only for fragments of @{text expr} that are not

  interpreted in the theory already.

  Equations given after @{keyword "where"} amend the morphism through

  which @{text expr} is interpreted.  This enables to map definitions

  from the interpreted locales to entities of @{text name}.  This

  feature is experimental.

  \item @{command "print_dependencies"}~@{text "expr"} is useful for

  understanding the effect of an interpretation of @{text "expr"}.  It

  lists all locale instances for which interpretations would be added

  to the current context.  Variant @{command

  "print_dependencies"}@{text "!"} prints all locale instances that

  would be considered for interpretation, and would be interpreted in

  an empty context (that is, without interpretations).

  \item @{command "print_interps"}~@{text "locale"} lists all

  interpretations of @{text "locale"} in the current theory or proof

  context, including those due to a combination of a @{command

  "interpretation"} or @{command "interpret"} and one or several

  @{command "sublocale"} declarations.

  \end{description}

  \begin{warn}

    Since attributes are applied to interpreted theorems,

    interpretation may modify the context of common proof tools, e.g.\

    the Simplifier or Classical Reasoner.  As the behavior of such

    tools is \emph{not} stable under interpretation morphisms, manual

    declarations might have to be added to the target context of the

    interpretation to revert such declarations.

  \end{warn}

  \begin{warn}

    An interpretation in a theory or proof context may subsume previous

    interpretations.  This happens if the same specification fragment

    is interpreted twice and the instantiation of the second

    interpretation is more general than the interpretation of the

    first.  The locale package does not attempt to remove subsumed

    interpretations.

  \end{warn}

*}

section {* Classes \label{sec:class} *}

text {*

  A class is a particular locale with \emph{exactly one} type variable

  @{text \<alpha>}.  Beyond the underlying locale, a corresponding type class

  is established which is interpreted logically as axiomatic type

  class \cite{Wenzel:1997:TPHOL} whose logical content are the

  assumptions of the locale.  Thus, classes provide the full

  generality of locales combined with the commodity of type classes

  (notably type-inference).  See \cite{isabelle-classes} for a short

  tutorial.

  \begin{matharray}{rcl}

    @{command_def "class"} & : & @{text "theory \<rightarrow> local_theory"} \\

    @{command_def "instantiation"} & : & @{text "theory \<rightarrow> local_theory"} \\

    @{command_def "instance"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command "instance"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

    @{command_def "subclass"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "print_classes"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

    @{command_def "class_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

    @{method_def intro_classes} & : & @{text method} \\

  \end{matharray}

  @{rail "

    @@{command class} class_spec @'begin'?

    ;

    class_spec: @{syntax name} '='

      ((@{syntax nameref} '+' (@{syntax context_elem}+)) |

        @{syntax nameref} | (@{syntax context_elem}+))      

    ;

    @@{command instantiation} (@{syntax nameref} + @'and') '::' @{syntax arity} @'begin'

    ;

    @@{command instance} (() | (@{syntax nameref} + @'and') '::' @{syntax arity} |

      @{syntax nameref} ('<' | '\<subseteq>') @{syntax nameref} )

    ;

    @@{command subclass} @{syntax target}? @{syntax nameref}

  "}

  \begin{description}

  \item @{command "class"}~@{text "c = superclasses + body"} defines

  a new class @{text c}, inheriting from @{text superclasses}.  This

  introduces a locale @{text c} with import of all locales @{text

  superclasses}.

  Any @{element "fixes"} in @{text body} are lifted to the global

  theory level (\emph{class operations} @{text "f\<^sub>1, \<dots>,

  f\<^sub>n"} of class @{text c}), mapping the local type parameter

  @{text \<alpha>} to a schematic type variable @{text "?\<alpha> :: c"}.

  Likewise, @{element "assumes"} in @{text body} are also lifted,

  mapping each local parameter @{text "f :: \<tau>[\<alpha>]"} to its

  corresponding global constant @{text "f :: \<tau>[?\<alpha> :: c]"}.  The

  corresponding introduction rule is provided as @{text

  c_class_axioms.intro}.  This rule should be rarely needed directly

  --- the @{method intro_classes} method takes care of the details of

  class membership proofs.

  \item @{command "instantiation"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s

  \<BEGIN>"} opens a theory target (cf.\ \secref{sec:target}) which

  allows to specify class operations @{text "f\<^sub>1, \<dots>, f\<^sub>n"} corresponding

  to sort @{text s} at the particular type instance @{text "(\<alpha>\<^sub>1 :: s\<^sub>1,

  \<dots>, \<alpha>\<^sub>n :: s\<^sub>n) t"}.  A plain @{command "instance"} command in the

  target body poses a goal stating these type arities.  The target is

  concluded by an @{command_ref (local) "end"} command.

  Note that a list of simultaneous type constructors may be given;

  this corresponds nicely to mutually recursive type definitions, e.g.\

  in Isabelle/HOL.

  \item @{command "instance"} in an instantiation target body sets

  up a goal stating the type arities claimed at the opening @{command

  "instantiation"}.  The proof would usually proceed by @{method

  intro_classes}, and then establish the characteristic theorems of

  the type classes involved.  After finishing the proof, the

  background theory will be augmented by the proven type arities.

  On the theory level, @{command "instance"}~@{text "t :: (s\<^sub>1, \<dots>,

  s\<^sub>n)s"} provides a convenient way to instantiate a type class with no

  need to specify operations: one can continue with the

  instantiation proof immediately.

  \item @{command "subclass"}~@{text c} in a class context for class

  @{text d} sets up a goal stating that class @{text c} is logically

  contained in class @{text d}.  After finishing the proof, class

  @{text d} is proven to be subclass @{text c} and the locale @{text

  c} is interpreted into @{text d} simultaneously.

  A weakend form of this is available through a further variant of

  @{command instance}:  @{command instance}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} opens

  a proof that class @{text "c\<^isub>2"} implies @{text "c\<^isub>1"} without reference

  to the underlying locales;  this is useful if the properties to prove

  the logical connection are not sufficent on the locale level but on

  the theory level.

  \item @{command "print_classes"} prints all classes in the current

  theory.

  \item @{command "class_deps"} visualizes all classes and their

  subclass relations as a Hasse diagram.

  \item @{method intro_classes} repeatedly expands all class

  introduction rules of this theory.  Note that this method usually

  needs not be named explicitly, as it is already included in the

  default proof step (e.g.\ of @{command "proof"}).  In particular,

  instantiation of trivial (syntactic) classes may be performed by a

  single ``@{command ".."}'' proof step.

  \end{description}

*}

subsection {* The class target *}

text {*

  %FIXME check

  A named context may refer to a locale (cf.\ \secref{sec:target}).

  If this locale is also a class @{text c}, apart from the common

  locale target behaviour the following happens.

  \begin{itemize}

  \item Local constant declarations @{text "g[\<alpha>]"} referring to the

  local type parameter @{text \<alpha>} and local parameters @{text "f[\<alpha>]"}

  are accompanied by theory-level constants @{text "g[?\<alpha> :: c]"}

  referring to theory-level class operations @{text "f[?\<alpha> :: c]"}.

  \item Local theorem bindings are lifted as are assumptions.

  \item Local syntax refers to local operations @{text "g[\<alpha>]"} and

  global operations @{text "g[?\<alpha> :: c]"} uniformly.  Type inference

  resolves ambiguities.  In rare cases, manual type annotations are

  needed.

  \end{itemize}

*}

subsection {* Co-regularity of type classes and arities *}

text {* The class relation together with the collection of

  type-constructor arities must obey the principle of

  \emph{co-regularity} as defined below.

  \medskip For the subsequent formulation of co-regularity we assume

  that the class relation is closed by transitivity and reflexivity.

  Moreover the collection of arities @{text "t :: (\<^vec>s)c"} is

  completed such that @{text "t :: (\<^vec>s)c"} and @{text "c \<subseteq> c'"}

  implies @{text "t :: (\<^vec>s)c'"} for all such declarations.

  Treating sorts as finite sets of classes (meaning the intersection),

  the class relation @{text "c\<^sub>1 \<subseteq> c\<^sub>2"} is extended to sorts as

  follows:

  \[

    @{text "s\<^sub>1 \<subseteq> s\<^sub>2 \<equiv> \<forall>c\<^sub>2 \<in> s\<^sub>2. \<exists>c\<^sub>1 \<in> s\<^sub>1. c\<^sub>1 \<subseteq> c\<^sub>2"}

  \]

  This relation on sorts is further extended to tuples of sorts (of

  the same length) in the component-wise way.

  \smallskip Co-regularity of the class relation together with the

  arities relation means:

  \[

    @{text "t :: (\<^vec>s\<^sub>1)c\<^sub>1 \<Longrightarrow> t :: (\<^vec>s\<^sub>2)c\<^sub>2 \<Longrightarrow> c\<^sub>1 \<subseteq> c\<^sub>2 \<Longrightarrow> \<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2"}

  \]

  \noindent for all such arities.  In other words, whenever the result

  classes of some type-constructor arities are related, then the

  argument sorts need to be related in the same way.

  \medskip Co-regularity is a very fundamental property of the

  order-sorted algebra of types.  For example, it entails principle

  types and most general unifiers, e.g.\ see \cite{nipkow-prehofer}.

*}

section {* Unrestricted overloading *}

text {*

  Isabelle/Pure's definitional schemes support certain forms of

  overloading (see \secref{sec:consts}).  Overloading means that a

  constant being declared as @{text "c :: \<alpha> decl"} may be

  defined separately on type instances

  @{text "c :: (\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n) t decl"}

  for each type constructor @{text t}.  At most occassions

  overloading will be used in a Haskell-like fashion together with

  type classes by means of @{command "instantiation"} (see

  \secref{sec:class}).  Sometimes low-level overloading is desirable.

  The @{command "overloading"} target provides a convenient view for

  end-users.

  \begin{matharray}{rcl}

    @{command_def "overloading"} & : & @{text "theory \<rightarrow> local_theory"} \\

  \end{matharray}

  @{rail "

    @@{command overloading} ( spec + ) @'begin'

    ;

    spec: @{syntax name} ( '==' | '\<equiv>' ) @{syntax term} ( '(' @'unchecked' ')' )?

  "}

  \begin{description}

  \item @{command "overloading"}~@{text "x\<^sub>1 \<equiv> c\<^sub>1 :: \<tau>\<^sub>1 \<AND> \<dots> x\<^sub>n \<equiv> c\<^sub>n :: \<tau>\<^sub>n \<BEGIN>"}

  opens a theory target (cf.\ \secref{sec:target}) which allows to

  specify constants with overloaded definitions.  These are identified

  by an explicitly given mapping from variable names @{text "x\<^sub>i"} to

  constants @{text "c\<^sub>i"} at particular type instances.  The

  definitions themselves are established using common specification

  tools, using the names @{text "x\<^sub>i"} as reference to the

  corresponding constants.  The target is concluded by @{command

  (local) "end"}.

  A @{text "(unchecked)"} option disables global dependency checks for

  the corresponding definition, which is occasionally useful for

  exotic overloading (see \secref{sec:consts} for a precise description).

  It is at the discretion of the user to avoid

  malformed theory specifications!

  \end{description}

*}

section {* Incorporating ML code \label{sec:ML} *}

text {*

  \begin{matharray}{rcl}

    @{command_def "use"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "ML"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "ML_prf"} & : & @{text "proof \<rightarrow> proof"} \\

    @{command_def "ML_val"} & : & @{text "any \<rightarrow>"} \\

    @{command_def "ML_command"} & : & @{text "any \<rightarrow>"} \\

    @{command_def "setup"} & : & @{text "theory \<rightarrow> theory"} \\

    @{command_def "local_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "attribute_setup"} & : & @{text "theory \<rightarrow> theory"} \\

  \end{matharray}

  @{rail "

    @@{command use} @{syntax name}

    ;

    (@@{command ML} | @@{command ML_prf} | @@{command ML_val} |

      @@{command ML_command} | @@{command setup} | @@{command local_setup}) @{syntax text}

    ;

    @@{command attribute_setup} @{syntax name} '=' @{syntax text} @{syntax text}?

  "}

  \begin{description}

  \item @{command "use"}~@{text "file"} reads and executes ML

  commands from @{text "file"}.  The current theory context is passed

  down to the ML toplevel and may be modified, using @{ML

  "Context.>>"} or derived ML commands.  The file name is checked with

  the @{keyword_ref "uses"} dependency declaration given in the theory

  header (see also \secref{sec:begin-thy}).

  Top-level ML bindings are stored within the (global or local) theory

  context.

  \item @{command "ML"}~@{text "text"} is similar to @{command "use"},

  but executes ML commands directly from the given @{text "text"}.

  Top-level ML bindings are stored within the (global or local) theory

  context.

  \item @{command "ML_prf"} is analogous to @{command "ML"} but works

  within a proof context.

  Top-level ML bindings are stored within the proof context in a

  purely sequential fashion, disregarding the nested proof structure.

  ML bindings introduced by @{command "ML_prf"} are discarded at the

  end of the proof.

  \item @{command "ML_val"} and @{command "ML_command"} are diagnostic

  versions of @{command "ML"}, which means that the context may not be

  updated.  @{command "ML_val"} echos the bindings produced at the ML

  toplevel, but @{command "ML_command"} is silent.

  \item @{command "setup"}~@{text "text"} changes the current theory

  context by applying @{text "text"}, which refers to an ML expression

  of type @{ML_type "theory -> theory"}.  This enables to initialize

  any object-logic specific tools and packages written in ML, for

  example.

  \item @{command "local_setup"} is similar to @{command "setup"} for

  a local theory context, and an ML expression of type @{ML_type

  "local_theory -> local_theory"}.  This allows to

  invoke local theory specification packages without going through

  concrete outer syntax, for example.

  \item @{command "attribute_setup"}~@{text "name = text description"}

  defines an attribute in the current theory.  The given @{text

  "text"} has to be an ML expression of type

  @{ML_type "attribute context_parser"}, cf.\ basic parsers defined in

  structure @{ML_struct Args} and @{ML_struct Attrib}.

  In principle, attributes can operate both on a given theorem and the

  implicit context, although in practice only one is modified and the

  other serves as parameter.  Here are examples for these two cases:

  \end{description}

*}

  attribute_setup my_rule = {*

    Attrib.thms >> (fn ths =>

      Thm.rule_attribute

        (fn context: Context.generic => fn th: thm =>

          let val th' = th OF ths

          in th' end)) *}

  attribute_setup my_declaration = {*

    Attrib.thms >> (fn ths =>

      Thm.declaration_attribute

        (fn th: thm => fn context: Context.generic =>

          let val context' = context

          in context' end)) *}

section {* Primitive specification elements *}

subsection {* Type classes and sorts \label{sec:classes} *}

text {*

  \begin{matharray}{rcll}

    @{command_def "classes"} & : & @{text "theory \<rightarrow> theory"} \\

    @{command_def "classrel"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\

    @{command_def "default_sort"} & : & @{text "local_theory \<rightarrow> local_theory"}

  \end{matharray}

  @{rail "

    @@{command classes} (@{syntax classdecl} +)

    ;

    @@{command classrel} (@{syntax nameref} ('<' | '\<subseteq>') @{syntax nameref} + @'and')

    ;

    @@{command default_sort} @{syntax sort}

  "}

  \begin{description}

  \item @{command "classes"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n"} declares class

  @{text c} to be a subclass of existing classes @{text "c\<^sub>1, \<dots>, c\<^sub>n"}.

  Isabelle implicitly maintains the transitive closure of the class

  hierarchy.  Cyclic class structures are not permitted.

  \item @{command "classrel"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} states subclass

  relations between existing classes @{text "c\<^sub>1"} and @{text "c\<^sub>2"}.

  This is done axiomatically!  The @{command_ref "subclass"} and

  @{command_ref "instance"} commands (see \secref{sec:class}) provide

  a way to introduce proven class relations.

  \item @{command "default_sort"}~@{text s} makes sort @{text s} the

  new default sort for any type variable that is given explicitly in

  the text, but lacks a sort constraint (wrt.\ the current context).

  Type variables generated by type inference are not affected.

  Usually the default sort is only changed when defining a new

  object-logic.  For example, the default sort in Isabelle/HOL is

  @{class type}, the class of all HOL types.

  When merging theories, the default sorts of the parents are

  logically intersected, i.e.\ the representations as lists of classes

  are joined.

  \end{description}

*}

subsection {* Types and type abbreviations \label{sec:types-pure} *}

text {*

  \begin{matharray}{rcll}

    @{command_def "type_synonym"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "typedecl"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

    @{command_def "arities"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\

  \end{matharray}

  @{rail "